§24.10 Arithmetic Properties

§24.10(i) Von Staudt–Clausen Theorem

Here and elsewhere in §24.10 the symbol denotes a prime number.

24.10.1

where the summation is over all such that divides . The denominator of is the product of all these primes .

24.10.2

where , and is an arbitrary integer such that . Here and elsewhere two rational numbers are congruent if the modulus divides the numerator of their difference.

§24.10(ii) Kummer Congruences

where .

valid when and , where is a fixed integer.

24.10.5

where is a prime and .

24.10.6

valid for fixed integers , and for all and such that .

§24.10(iii) Voronoi’s Congruence

Let , with and relatively prime and . Then

24.10.7

where and are integers, with relatively prime to .

For historical notes, generalizations, and applications, see Porubský (1998).

§24.10(iv) Factors

With as in §24.10(iii)

24.10.8

valid for fixed integers , and for all such that and .

24.10.9

valid for fixed integers and for all such that .